Summary: J. Phys. A: Math. Gen. 32 (1999) 7049­7056. Printed in the UK PII: S0305-4470(99)04810-6
Glass transition in self-organizing cellular patterns
Tomaso Aste and David Sherrington
INFM, C so Perrone 24, Genova Italy and LDFC, 3 rue de l'Universit´e, Strasbourg, France
Department of Physics, University of Oxford, Theoretical Physics, 1 Keble Road, Oxford OX1
3NP, UK
Received 3 June 1999
Abstract. We have considered the dynamical evolution of cellular patterns controlled by a
stochastic Glauber process determined by the deviations of local cell topology from that of a
crystalline structure. Above a critical temperature evolution is towards a common equilibrium state
from any initial configuration, but beneath this temperature there is a dynamical phase transition,
with a start from a quasi-random state leading to non-equilibrium glassy freezing whereas an
ordered start rests almost unchanged. A temporal persistence function decays exponentially in
the high-temperature equilibrating state but has a characteristic slow non-equilibrium ageing-like
behaviour in the low-temperature glassy phase.
The hexagonal tiling is the best partition of the plane in equal cells. It solves both the packing
and the covering problems. It is the space-filling assembly of equal cells with the minimal
interfacial extension.
It is regular, perfect and beautiful. Surprisingly, however, it is never realized in natural
biological tissues where a relevant amount of disorder is always present [1, 2]. This is